Jacobian of solutions to the conductivity equation in limited view

TL;DR

This paper investigates conditions ensuring the Jacobian of solutions to the conductivity equation remains non-zero in a limited view setting, crucial for hybrid inverse problems like Acousto-Electric Tomography, and explores numerical reconstruction impacts.

Contribution

It provides sufficient boundary conditions for non-vanishing Jacobians in limited view scenarios, including discontinuous boundary functions, and analyzes their effect on conductivity reconstruction.

Findings

Non-vanishing Jacobian conditions are established for limited boundary control.

Discontinuous boundary functions can be incorporated using weighted Sobolev spaces.

Numerical reconstructions show the impact of limited view on solution quality.

Abstract

The aim of hybrid inverse problems such as Acousto-Electric Tomography or Current Density Imaging is the reconstruction of the electrical conductivity in a domain that can only be accessed from its exterior. In the inversion procedure, the solutions to the conductivity equation play a central role. In particular, it is important that the Jacobian of the solutions is non-vanishing. In the present paper we address a two-dimensional limited view setting, where only a part of the boundary of the domain can be controlled by a non-zero Dirichlet condition, while on the remaining boundary there is a zero Dirichlet condition. For this setting, we propose sufficient conditions on the boundary functions so that the Jacobian of the corresponding solutions is non-vanishing. In that regard we allow for discontinuous boundary functions, which requires the use of solutions in weighted Sobolev spaces. We implement the procedure of reconstructing a conductivity from power density data numerically and investigate how this limited view setting affects the Jacobian and the quality of the reconstructions.